These are some mathematics investigations I have pursued over the years. They may be of some interest to teachers, students, or hobbyists. I try to convey a conceptual understanding, usually without rigorous proof. Some of the lessons are accompanied by questions and suggestions for extensions.

cuero / malla

Importado

Caucho suela

Plus Fitting Sistema - Dos extraíbles Camas para agregar y Doble Profundidad

Drilex Forro con AGEIS Escudo microbio para el olor y la mancha Control

Suave, Foam-PaddFlared, Goma ligera Suela con EVA Cuña entresuela - Extraíble, poliuretano Plantilla con talón Cojín y Máximo - Arco construido Soporte

Completo Contacto Rocker Lengua de fondo y Collar - Acomoda Presctibed Orthotics

Tallas: 5-13 - Anchos: AA, B, D, 2E, 4E

Some of the lessons are accompanied by questions and suggestions for extensions.

In my pursuit of a teaching career, I was told to justify the study of each concept by establishing its relevance to the students’ lives. Sorry, but I still have trouble buying into that one. Fortunately, poetry and music are rarely put to that same test. As I glance at the list below, I must concede that it would be difficult to convert any of the lessons into food, shelter, or money. These are things that interested me, and now I understand them better. If there is a reward, it is the fact that every time I do this I get better at figuring things out.

I occasionally get email from people seeking permission to use my work. Feel free to cite, quote, edit, reproduce, or distribute anything on this site. Do not worry about putting my name on it, but please do not put your own name on it.

Many of these presentations use the Geometer’s Sketchpad, but only one of them actually requires it. If you do not have the software, let me suggest that you follow this link to Key Curriculum Press and look into it.

Paul Kunkel

March 16, 2016

Updated the animations on Hanging With Galileo.

October 23, 2015

Updated the animations on Reuleaux Triangle.

October 9, 2015

Updated the animations on the Vernier Scale page of the Geometry of Surveying.

October 8, 2015

Drew Shoes Toscana Drew Diabético Terapéutico Mujer Profundidad Extra Zapatos Extra Negro de Cuero/ Malla Con Cordones Negro/ Combo bb639e2Updated the animations on Linkages.

October 4, 2015

Updated the animations on On the Square.

September 25, 2015

Updated the animations on Tangent Circles.

September 23, 2015

Updated the lead page of Conic Sections and Construction.

The page it replaced had java animations that were not being treated well by modern browsers. New images were generated with the developmental Web Sketchpad, which is producing very satisfying results. This appears to be the future, so expect more of my pages to be similarly updated soon.

February 26, 2015

Added Book VII to the Conics of Apollonius.

This is the final surviving book of the treatise.

October 22, 2014

Added Book VI to the Conics of Apollonius.

June 10, 2014

Added Book V to the Conics of Apollonius.

Includes GSP-3 files

Includes GSP-4 files

Includes GSP-5 files
Contents:

We are given two fixed points in a vertical plane. A particle starts from rest at one of the points and travels to the other under its own weight. Find the path that the particle must follow in order to reach its destination in the briefest time.

This is a famous problem in the calculus of variations. Nothing new is presented here, but the explanation is more thorough than most.

Buffon’s Needle

An old probability exercise is aided by an interactive Sketchpad file. By manipulating the sketch, an intuitive understanding may be gained, even if the student has had no introduction to calculus.

Drew Shoes Toscana Drew Diabético Terapéutico Mujer Profundidad Extra Zapatos Extra Negro de Cuero/ Malla Con Cordones Negro/ Combo bb639e2

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

Diadora B. Original Sneaker para hombre Almendra Beige , Zapatillas para mujer Del Rey TOMS Negro, cepillado, tejido, , Converse Chuck Taylor All Star Core Ox Casi negro Zapatillas de tenis adidas Asmc Barricade 2017 para mujer Blanco / Universo / Electricidad K-Swiss Women s Tubes Millennia CMF Cross Trainer Angel Wing / Blanco Zapatos atléticos para diabéticos Meghan Turquoise de Dr. Comfort para mujer Púrpura , Zapatilla de running Titan II para hombre Diadora Blue Jay / Paloma Gray Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Pewter / Calf Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco azul , Dr. Zen Jordan Comfort Profundidad extra terapéutica para mujer Cuero para calzado / malla con cordones Gris , Zapatillas de deporte adidas by Stella McCartney Ultraboost Parley Sneakers de adidas by Stella McCartney Res. Alta Azul S18 / Núcleo Blanco / Oscuro Callistos 07 , ASICS Tiger Womens Gel-Lyte V Skyway / Skyway , Phenma femenino Phenom Satin EP Puma Black / Puma Black , Zapatillas de tenis ASICS Gel-Solution Speed 3 para mujer Plata blanca , Zapatillas para mujer Del Rey TOMS Tela escocesa negra / blanca Zapatilla de running Titan II para hombre Diadora Khaki / Estate Blue Zapatillas para mujer Babolat Propulse BPM rosa Diadora Unisex N9000 III Blanco / alto Lakai Daly Skate Shoe Textil de Borgoña , Zapatillas de tenis ASICS Gel-Challenger 11 para mujer Plata blanca , Diadora Unisex I.C. Pompeian Red / Nautical Blue New Balance Wrl247hk para mujeres Chocolate Zapatilla de deporte de gamuza bala para mujer Gola Azul cielo / blanco roto Columbia Womens Santiam Flip Gray Ash / Pacific Rim Zapatillas para mujer Del Rey TOMS Lona negra de poliéster , APL: zapatillas de deporte Athletic Propulsion Labs para mujer Techloom Pro Cachemira de trigo Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Crema / Combo Zapatilla de deporte de gamuza bala para mujer Gola Azul marino / grisáceo , Zapatillas Pallaphoenix CVS PRT Sneaker de Palladium para mujer Blanco ,

trigonometry, and probability. Fair warning: The discussion revolves about
a figure that appears only when the sketch is animated. Without Geometer’s
Sketchpad, this one will make little sense at all.
Conic Sections and Construction

My emphasis is on geometric properties of the conic sections, particulary understanding that has been handed down to us in ancient texts. There also are some challenging constructions.

The Geometry of Surveying

This used to be my job, and I enjoyed studying all of the geometrical concepts that were part of the science of land surveying. In this article I discuss the instruments of the profession, their operating principles, their error, and their calibration.

Hanging with Galileo

An investigation into the catenary, the curve formed by a chain suspended at both ends. It also extends into the curve of the main cables on a suspension bridge, which, interestingly, is not a catenary.

Inversion Geometry KEEN Hombres Newport H2 Sandal Imán / Capuchina Drew Womens Cascade Sand Nubuck BZees Sandalia con cuña de beso para mujer Blanco , Zapatillas de running Spira Phoenix para mujer con resortes Blk / Sky , Zapatillas de running Saucony Women s Freedom ISO Azul / Citron , Sandalia de deslizamiento ultra Comfort NIKE para mujer Pure Platinum / Wolf Gray Chaco Mujer ZX1 Classic Sport Sandalia Origami Black , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Azul Púrpura / Negro / Flash Coral , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Deep Porcelain Blue / Overcast Zapatillas de running New Balance 990V3 para mujer Armada SOREL Womens Joanie Wrap Sandal Negro Zapato de running ASICS Gel-Quantum 360 cm para mujer Punto medio gris / glaciar / coral , Under Armour Men s Charged Bandit 3 Stealth Gray (100) / Negro , Nueve West Womens Xaeden Cuero negro G.H. Negro SOREL Womens Joanie Wrap Sandal Sáhara Prop & eacute; t Zapatillas para caminar TravelFit para mujer Gris oscuro Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Equipaje Tan , Sandalia plana Dionne de Easy Street para mujer Equipaje Sandalia Corey de Dolce Vita para mujer Rafia negra múltiple Zapatillas de skate casual unisex Heathrow Prestige para hombre de DC Negro / Marrón / Blanco , Botines metálicos A-Game para mujer Boombah - 6 opciones de color - Varios tamaños Negro / real , Haflinger Women s Phantasy Slip on Slipper Gris , Zapatilla de gamuza Lynneah de Nine West para mujer Dark Natural Suede Pantuflas Matilda de Katy Perry para mujer Blush desnuda , Tommy Hilfiger Mujer Balbina Oxford Gris pardo BareTraps BT Ricarda tobillera para mujer Negro Timberland Heritage s Women s Noreen - Bote con tres ojos para barcos marrón , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Dark Navy / Drizzle Grey Slub Chambray Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Blanco ,

If you have never studied inversion geometry, you can start here. The elementary concepts are explained. This page is a quick reference for the other lessons that employ inversion geometry.

Linkages

Draw a line segment. Did you use a straightedge? How do you know it was straight?

Java simulations are used to investigate a nineteenth century mechanics problem.

On the Square

The challenge: Given four arbitrary, unique points, *A*, *B*,
*C*, and *D*, construct one line through each point such that the four lines form a square.

Orthocenter What is so significant about the orthocenter of a triangle? Why does it even have a name? This investigation was prepared for a geometry teachers’ workshop. It includes a lesson handout in Word format. This lesson was written for use in a computer lab with the Geometer’s Sketchpad, but with a little editing it should work just as well with Cabri or Cinderella.

The Planimeter

A planimeter is a drafting instrument used to measure area. Its mechanism is quite simple, but understanding why it works is more elusive.

Platonic Solids

Find the inradius, circumradius, dihedral angle, surface area, and volume of all of the Platonic solids. There is also some information about some of their other interesting properties.

Reuleaux Triangle

It is not a circle, but it has constant width. Study the properties of this figure with interactive animated sketches.

Rungs and Vineyards

How is that for a descriptive title? Here are investigations of two interesting visual effects seen from a moving automobile. It includes two interactive perspective drawings, and a discussion of the cylindrical and planar projections used to create them.

The Sliding Triangle

An interesting accidental discovery – Investigate ellipses by tracing triangle transformations.

Table Patterns

Explore geometric patterns formed in a table of modular multiplication. This one requires some elementary knowledge of congruence classes and modular arithmetic.

Tangent Circles

Three circles are given. How many circles can be constructed tangent to all three? How are they constructed? This lesson includes ten animations covering all of the special cases. Some of them are very complex. A little knowledge of circle inversions would help.

Volume of a Torus Do you already know the volume of a torus? Then humor me. I just figured Drew Shoes Toscana Drew Diabético Terapéutico Mujer Profundidad Extra Zapatos Extra Negro de Cuero/ Malla Con Cordones Negro/ Combo bb639e2 it out, and I got a chance to try out this new graphics software.

Resources

The Geometer’s Sketchpad Workshop

It is plain to see that I am very fond of Sketchpad. This page was created after the release of version 4. The idea is to have a place to exchange ideas with other users, especially those who are using the advanced features. Here are some custom tools and some information on Java Sketchpad.

Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge
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and compass constructions are described and illustrated. The original version,
in Word format, can be downloaded and distributed.

Last update: March 16, 2016 ... Paul Kunkel

For email to reach me, the word